# Coclass of a nilpotent group

## Definition

The **coclass** of a finite nilpotent group is defined as the difference between the sum of exponents of prime divisors, of its order and its nilpotence class. The term coclass is typically used for groups of prime power order, for which it is simply the exponent of the prime in its order, minus the nilpotence class.

## Related terms

- Maximal class group is a group of coclass one. We cannot have a nontrivial group of coclass zero.

## Importance

Coclass has been used as a primary classification invariant in attempted classifications of -groups.